Transcript Hough Transform

Hough Transform
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Hough Transform
A Technique to Isolate Features of
a Particular Shape within an Image
The classical Hough transform is most commonly
used for the detection of regular curves such as:
lines, circles, ellipses, etc
Because:
The desired features be specified in some parametric form
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Voting schemes
 Let each feature vote for all the models that are compatible with it
 Hopefully the noise features will not vote consistently for any
single model
 Missing data doesn’t matter as long as there are enough features
remaining to agree on a good model
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Properties
 Can tolerate noise and gaps in edge image
 Look for solutions in a parameter space
 Classical Hough transform
 Detect simple shape
 Line detection
 Circle detection
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Hough transform
 An early type of voting scheme
 General outline:
 Discretize parameter space into bins
 For each feature point in the image, put a vote in every bin in the
parameter space that could have generated this point
 Find bins that have the most votes
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Hough Transform
• Instead of going directly for the solution (as in template matching), we
can use a voting scheme.
• Each possible point (edge) votes in the parameter space of a line.
• A line can be specified uniquely by two parameters m and n.
y = m.x + n
• Parameter space of 2 dimensions (m and n).
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‫شنبه زاده‬
Finding lines in an image
y
b
b0
x
image space
m0
m
Hough space
A line in the image corresponds to a point in Hough space
Connection between image (x,y) and Hough (m,b) spaces
• A line in the image corresponds to a point in Hough space
• To go from image space to Hough space:
- given a set of points (x,y), find all (m,b) such that y = mx + b
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 Where is the line that contains both (x0, y0) and (x1, y1)?
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 Where is the line that contains both (x0, y0) and (x1, y1)?
 It is the intersection of the lines b = –x0m + y0 and b = –
x1m + y1
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y  mx  c
Equation of Line:
Find:
(m, c)
Consider point:
y
y  mx  c
( xi , yi )
( xi , yi )
yi  mxi  c or c   xi m  yi
Image Space
Parameter space also called Hough Space
x
m
(m, c)
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Parameter Space
c
 Algorithm:
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Quantize Parameter Space
(m, c)
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Create Accumulator Array
A(m, c)
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Set
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For each image edge
A(m, c)  0 m, c
( xi , yi )
(m, c)
increment:
A(m, c)  A(m, c)  1
Parameter Space
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If
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(m, c)
lies on the line:
Find local maxima in
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2
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c   xi m  yi
A(m, c)
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Problem with m-n parameter space.
 Both m and n can get very large [-∞, ∞].
How do we address this problem?
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‫شنبه زاده‬
An alternative parameter space is polar coordinates. We use (ρ,θ)
space to represent a line.
ρ = x.cos(θ) + y.sin(θ)
Points in picture
 sinusoids in parameter space
Points in parameter space

lines in picture
Sinusoids corresponding to co-linear points intersect at an unique point
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Example:
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Line: 0.6x + 0.4y = 2.4
Sinusoids intersect at:ρ = 2.4, θ = 0.9273
Let ρ= [−R, R] and θ= [0,pi )
For each edge point (xi , yi ),
calculate:
θ ‘ = xi cos θ ‘+ yi sin θ
Accumulator:
A(ρ, θ) = A(ρ, θ) + 1
Threshold the accumulator values to
get parameters for detected lines
Threshold at A(ρ, θ) = 30
Threshold at A(ρ, θ) = 20
Threshold at A(ρ, θ) = 15
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‫شنبه زاده‬
Example
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Hough Transform Example with breaks in lines
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Hough Transform Example with 1% salt and pepper noise
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The Hough transform of the line is simply:
a point at coordinate (ρ,θ)
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A family of lines passing through a common point
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The Hough transform of the family of curves passing through the
three points results in the set of three parametric curves.
These three curves cross at a single point corresponding to the
dashed line passing through the collinear points.
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Example
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Tokens
Theta = 45º = 0.785 rad
r = (1√2) / 2 = 0.707
Brightest point = 20 votes
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Votes
Theta: 0 to 3.14 (rad)
Brightest point = 6 votes
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Noise Lowers the Peaks
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Noise Increases the Votes in Spurious Accumulator Elements
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( e) Straight dashed line
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( f) Hough transform of dashed line
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Advantages
• Edges need not be connected
• Complete object need not be visible
• Key Idea: Edges VOTE for the possible model
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Problems with HT
1)Exponential time and memory increase with the
increase of the parameter numbers.
2) Sampling of parameter space is difficult.
3) Similar shapes might produce peaks in the Hough Space
(wide ellipses might interfere with lines)
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A city scene where the buildings are obstructed in fog
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To find the true edges of the buildings
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Hough transform can detect some of the straight lines
representing building edges within the obstructed region
The histogram equalized
accumulator space
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‫شنبه زاده‬
De-Houghed image
set the relative threshold to 70%
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Relative Threshold 50%
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Street information is not available
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Hough line detector is able to recover some of this information
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Hough Transform By MATLAB
I = imread('circuit.tif');
BW = edge(I,'canny');
[H,T,R] = hough(BW);
P = houghpeaks(H,12,'threshold',ceil(0.3*max(H(:))));
lines = houghlines(BW,T,R,P,'FillGap',5,'MinLength',7);
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Hough transform for circles
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Hough transform for circles
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Implementation of the Hough transform
for circles
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‫شنبه زاده‬
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‫آلگوريتم زير را از صفحه ‪ 206‬کتاب استخراج ويژگی های تصوير مطالعه نموده و‬
‫برای يک تصوير چهره عنبيه را مشخص سازيد‪.‬‬
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Hough transform for ellipses
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Circles are very important in shape detection since many objects have a circular
shape. However, because of the camera’s viewpoint, circles do not always look like
circles in images.
We can define the mapping between the circle and an ellipse by a similarity
transformation.
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